# Ecological Dynamic Programming Optimization in Python

It’s been awhile since I’ve posted anything. I’ll use the excuse that I’ve been busy, but mostly I just forget. Regardless, I’m learning how to do Dynamic Programming Optimization (DPO), which sounds more complex than it is. The reason for this is that DPO allows us to simulate the behavior of individuals who make decisions based on current patch quality.  DPO is an exciting tool that forms the foundation of individual-based models, which allow us to assess community and ecosystem dynamics as emergent properties of individual decisions based on well-grounded, physiological principles (hence my interest).The underlying idea, as I understand it, is that individuals assess their future reproductive output prior to making a decision (I’m interested in optimal foraging, so I’ll put this in the context of foraging). We can model how individuals make decisions over the course of a lifetime, and track each individual, which then allows us to make quantitative statements about populations, communities, or ecosystems.

This all sounds complicated, and it can be difficult. So it’s easiest to jump right in with an example. Here’s the code, along with a basic explanation of what’s going on, for the very first toy example found in “Dynamic State Variable Models in Ecology” by Clark and Mangel. In the book, they give the computer code in TRUEBASIC, but… in all honesty.. no one I know uses that. I could use R, but we all know my feelings on that. So here’s their toy model programmed and annotated in Python.

Background

Suppose we’re interested in the behavior of fish and how they choose to distribute themselves among three different patches. Two patches are foraging patches and one is a reproduction patch. The first thing we need to do is make an objective fitness function F. In this example, F relates body size to reproductive output. It can be thought of as what the organism believes about its final fitness given a particular body size. Let’s make it increasing, but asymptotic, with body mass:

$F = \frac{A(x-x_c)}{x-x_c+x_0}$

Here, A is sort of a saturation constant beyond which fitness increases minimally with body size, $x_c$ is the body size at which mortality occurs (0), and $x_0$ is the initial body size. Lets set $x_c=0$, $A=60$ and $x_0=0.25*x_{max}$, where $x_{max}=30$ is the individuals maximum possible body size. You have to set the ceiling on body size otherwise organisms grow without bounds.

OK so that’s what the organisms “believes” about its fitness at the end of a season given a specific body mass. The question is: How does the organism forage optimally to maximize fitness at the end of the its lifetime?

The obvious answer would be to simulate a foraging decision at every time step moving forward, and then have it decide again at the new time step, etc. This is computationally expensive, so to circumvent this we work backwards from the end of the season. This saves time because there are far fewer paths to a known destination than there are to an unknown one (essentially).

So we imagine that, at the final time step $t_f$, the organism’s fitness is given by F for any given body mass.

import numpy as np
import matplotlib.pyplot as pet
import pandas as pd

n_bodysize = 31 # 31 body sizes, from 0-30
n_time = 20 # 20 time steps, 1-20
n_patch = 3, # 3 foraging patches

# make a container for body size (0-30) and time (1-20)
F = np.zeros((n_bodysize, n_time))

# make the function F
x_crit = 0
x_max = 30
acap = 60
x_0 = 0.25*x_max
t_max = 20

# calculate organism fitness at the final time point
for x in range(x_crit, x_max+1):
F[x,-1] = acap*(x-x_crit)/(x-x_crit+x_0)
[/code]

Now that we know fitness at the final time point, we can iterate through backwards (called backwards simulation) to decide what the optimal strategy is to achieve each body mass. To determine that, we need to know the fitness in each patch. Let’s start with the two foraging patches, Patch 1 and Patch 2. We need to know four things about each patch: (1) the mortality rate in each patch (m), (2) the probability of finding food in each patch (p), (3) the metabolic cost of visiting a patch (a), and (4) the gain in body mass if food is successfully found (y). For these two patches, let:

Patch 1: m=0.01, p=0.2, a=1, y=2

Patch 2: m=0.2, p=0.5, a=1, y=4

Right away we can see that Patch 2 is high-risk, high reward compared to Patch 1. In each patch, we calculate the next body size given that an animal does (x‘) or does not (x”) find food:

$x' = x-a_i+y_i$

$x'' = x-a_i$

Those are simple equations. Body size is the current body size minus the metabolic cost of foraging in the patch and, if successful, the energy gain from foraging. Great. Now we can calculate the expected fitness of each patch as the weighted average of F‘ and F” given the probability of finding food, all times the probability of actually surviving. For these two patches, we make a fitness function (V):

$V_i = (1 - m_i)*[p_i*F_{t+1}(x') + (1-p_i)*F'_{t+1}(x'')]$

The reproductive patch is different. There is no foraging that occurs in the reproductive patch. Instead, if the organism is above a critical mass $x_{rep}$, then it devotes all excess energy to reproduction to a limit  (c=4). If the organism is below the reproductive threshold and still visits the foraging patch, it simply loses mass (unless it dies).

OK this is all kind of complicated, so let’s step through it. We know what fitness is at the final time step because of F. So let’s step back one time step. At this penultimate time step, we go through every body mass and calculate fitness for each patch. Let’s do an example. If x=15, then we need to know fitness in Patch 1, Patch 2, and Patch 3. For Patches 1 and 2, we need to know the weight gain if successful and the weight gain if unsuccessful.

$x' = max(15-a_1+y_1, x_{max})$

$x' = 15-1+2 = 16$

$x''=min(15-a_1, x_{c})$

$x'' = 15-1 = 14$

The min and max functions here just make sure our organism doesn’t grow without limit and dies if metabolic cost exceeds body mass. So these are now the two potential outcomes of foraging in Patch 1 given x=15. The expected fitness of these two body masses is given as F(16) and F(14). Plug all these values into equation V for Patch 1 to get the expected fitness of Patch 1 at a body size of 15. We then take the maximum for all three Patches, save whichever Patch corresponds to that as the optimal foraging decision, and then save as the fitness for body size 15 at that time step. So at body size 15, for Patch 1 is  39, for Patch 2 is 38, and for Patch 3 is 37.6, so the individual will foraging in Patch 1 and now fitness for body size 15 at this time step is 39.

Repeat this procedure for every possible body size, and you’ll get the fitness for every body size at the second to last time step as well as the optimal foraging patch for every body size at that time.

Then, step backwards in time. Repeat this whole procedure, except now the value for each body mass doesn’t come from the equation F, but comes from the fitness we just calculated for each body mass. So for example, if an organism’s foraging decisions at this time step lead it to a body mass of 15, then F is now 39. Again, repeat this for every body mass, and then step back, etc.

Here’s the full Python code for how this is done:

import numpy as np
import matplotlib.pyplot as plt
import pandas as pd
import seaborn as sns

n_bodysize = 31 # 31 body sizes, 0 - 30
n_time = 20 # 20 time steps, 1 - 20
n_patch = 3 # number of patches

#%% CONATINERS
# make a container for body size (0-30) and time (1-20)
F = np.zeros((n_bodysize, n_time))
# make a container for three patches, each with body size (0-30) and time (1-20)
Vi = np.zeros((n_patch, n_bodysize, n_time))
# make a container for the optimal decision
d = np.zeros((n_bodysize, n_time))

# make a container for the mortality rates in each patch
m = np.zeros(n_patch)
# make a container for the probability of finding food in each path
p = np.zeros(n_patch)
# make a container for the metabolic cost
a = np.zeros(n_patch)
# make a container for food gain in each patch
y = np.zeros(n_patch)
# make a container for reproductive output in each patch
c = np.zeros(n_patch)

#%% CONDITIONS
x_crit = 0
x_max = 30
t_max = 20
x_rep = 4

#%% PARAMETERS
m[0] = 0.01; m[1] = 0.05; m[2] = 0.02
p[0] = 0.2; p[1] = 0.5; p[2] = 0
a[0] = 1; a[1] = 1; a[2] = 1
y[0] = 2; y[1] = 4; y[2] = 0
c[0] = 0; c[1] = 0; c[2] = 4

#%% END CONDITION
acap = 60
x_0 = 0.25*x_max

# Calculate Fitness for every body mass at the final time step
for x in range(x_crit, x_max+1):
F[x,-1] = acap*(x-x_crit)/(x-x_crit+x_0) # maximum reproductive output for each body size at the final time

#%% SOLVER
for t in range(0, t_max-1)[::-1]: # for every time step, working backward in time #print t
for x in range(x_crit+1, x_max+1): # iterate over every body size # print x
for patch in range(0, 3): # for every patch
if patch in [0,1]: # if in a foraging patch
xp = x-a[patch]+y[patch] # the updated body size
xp = min(xp, x_max) # constraint on max size
xpp = int(x-a[patch]) # updated body size if no food
Vi[patch,x,t] = (1 - m[patch])*(p[patch]*F[int(xp),t+1] + (1-p[patch])*F[xpp, t+1]) # Calculate expected fitness for foraging in that patch
else:
if x < x_rep: # in a reproduction patch
xp = max(x-a[patch], x_crit)
Vi[patch, x, t] = (1-m[patch])*F[int(xp), t+1]
else:
fitness_increment = min(x-x_rep, c[patch]) # resources devoted to reproduction
xp = max(x-a[patch]-fitness_increment, x_crit) # new body size is body size minus metabolism minus reproduction
Vi[patch, x, t] = fitness_increment + (1-m[patch])*F[int(xp),t+1]
vmax = max(Vi[0,x,t], Vi[1,x,t])
vmax = max(vmax, Vi[2,x,t])
if vmax==Vi[0,x,t]:
d[x,t] = 1
elif vmax==Vi[1,x,t]:
d[x,t] = 2
elif vmax==Vi[2,x,t]:
d[x,t] = 3
F[x,t] = max # the expected fitness at this time step for this body mass
[/code]

This model doesn’t really track individual behavior. What it does is provides an optimal decision for every time and every body mass. So we know what, say, an individual should do if it finds itself small towards the end of its life, or if it finds itself large at the beginning:

T,X = np.meshgrid(range(1, t_max+1), range(x_crit, x_max+1))
df = pd.DataFrame({'X': X.ravel(), 'T': T.ravel(), 'd': d.ravel()})
df = df.pivot('X', 'T', 'd')
sns.heatmap(df, vmin=1, vmax=3, annot=True, cbar_kws={'ticks': [1,2,3]})
plt.gca().invert_yaxis()
plt.show()
[/code]

And that’s that!! To be honest, I still haven’t wrapped my head around this fully. I wrote this blog post in part to make myself think harder about what was going on, rather than just regurgitating code from the book.

# My Ideal Python Setup for Statistical Computing

I’m moving more and more towards Python only (if I’m not there already). So I’ve spent a good deal of time getting the ideal Python IDE setup going. One of the biggest reasons I was slow to move away from R is that R has the excellent RStudio IDE. Python has Spyder, which is comparable, but seems sluggish compared to RStudio. I’ve tried PyCharm, which works well, but I had issues with their interactive interpreter running my STAN models.

A friend pointed me towards SublimeText 3, and I have to say that it’s everything I wanted. The text editor is slick, fast, and has lots of great functions. But more than that, the add-ons are really what make Sublime shine

• Side Bar Enhancements: This extends the side-bar project organizer, allowing you to add folders and files, delete things, copy paths, etc. A must have.
• SublimeREPL: Adds interactive interpreters for an enormous number of languages, both R and Python included. Impossible to work without.
•  Anaconda: An AMAZING package that extends Sublime by offering live Python linting to make sure my code isn’t screwed up, PEP8 formatters for those of you who like such things, and built in documentation and code retrieval, for those times you’ve forgotten how the function works. Another must have.
• SublimeGIT: For working with github straight from Sublime. Great if you’re doing any sort of module building.
• Origami: A new way to split layouts and organize your screen. Not essential, but helpful
• Bracket Highlighter: Helpful for seeing just what set of parentheses I’m working in.

Sublime and all of these packages are also incredibly customizable, you can make them work and look however you want. I’ve spent a few days customizing my setup and I think its fairly solid. Here are my preferences:

For the main Sublime, I modified the scrolling map, turned off autocomplete (which I find annoying but can still access with Ctrl+space, adjusted the carat so I could actually see it, changed the font, and a few other odds and ends.

{
"always_show_minimap_viewport": true,
"auto_complete": false,
"bold_folder_labels": true,
"caret_style": &amp;quot;phase&amp;quot;,
"color_scheme": "Packages/Theme - Flatland/Flatland Dark.tmTheme",
"draw_minimap_border": true,
"font_face": "Deja San Mono",
"font_size": 14,
"highlight_line": true,
"highlight_modified_tabs": true,
"ignored_packages":
[
"Vintage";
],
"preview_on_click": false,
"spell_check": true,
"wide_caret": true,
}

For Bracket Highlighter, I changed the style of the highlight:

{
"high_visibility_enabled_by_default": true,
"high_visibility_style": "thin_underline",
"high_visibility_color": "__default__",
}

For Side-Bar Enhancements, I’ve modified the ‘Open With’ options. For Anaconda, I changed a few small things and turned off PEP8 linting, which I hate. I don’t hate linting nor PEP8, but I don’t have much use for PEP8 linting constantly telling me that I put a space somewhere inappropriate.

{
"complete_parameters": true,
"complete_all_parameters": false,
"anaconda_linter_mark_style": "outline",
"pep8": false,
"anaconda_gutter_theme": "basic",
"anaconda_linter_delay": 0.5,
}

I also installed the Flatland Theme to make it pretty. Here is the end result, also showing the Anaconda documentation viewer that I find so awesome:

I also now use Sublime for all of my R, knitr, and LaTeX work as well. In all, it’s a pretty phenomenal editor that can do everything I need it to and combines at least four separate applications into one (TextWrangler, Spyder, RStudio, TexShop). Now, some day I’ll be able to afford the $70 to turn off that reminder that I haven’t paid (and$15 for LaTeXing).

UPDATE

I forgot to mention snippets. You can create snippets in Sublime that are shortcuts for longer code. For example, I heavily customize my graphs in the same way every time. Instead of typing all the code, I can now just type tplt followed by a tab and I automatically get:

f, ax = plt.subplots()
ax.plot()
#ax.set_ylim([ , ])
#ax.set_xlim([ , ])
ax.set_ylabel(&amp;quot;ylab&amp;quot;)
ax.set_xlabel(&amp;quot;xlab&amp;quot;)
ax.spines['top'].set_visible(False)
ax.spines['right'].set_visible(False)
ax.spines['bottom'].set_position(('outward', 10))
#ax.spines['bottom'].set_bounds()
ax.spines['left'].set_position(('outward', 10))
#ax.spines['left'].set_bounds()
ax.yaxis.set_ticks_position('left')
ax.xaxis.set_ticks_position('bottom')
plt.savefig(,bbox_inches = 'tight')
plt.show()

Great if you rewrite the same code many times.

# PyStan: A Second Intermediate Tutorial of Bayesian Analysis in Python

I promised a while ago that I’d give a more advanced tutorial of using PySTAN and Python to fit a Bayesian hierarchical model. Well, I’ve been waiting for a while because the paper was in review and then in print. Now, it’s out and I’m super excited! My first pure Python paper, using Python for all data manipulation, analysis, and plotting.

The question was whether temperature affects herbivory by insects in any predictable way. I gathered as many insect species as I could and fed them whatever they ate at multiple temperatures. Check the article for more detail, but the idea was to fit a curve to all 21 herbivore-plant pairs as well as to estimate the overall effect of temperature. We also suspected (incorrectly as it turns out) that plant nutritional quality might be a good predictor of the shape of these curves, so we included that as a group-level predictor.

Anyway, here’s the code, complete with STAN model, posterior manipulations, and some plotting. First, here’s the actual STAN model. NOTE: a lot of data manipulation and whatnot is missing. The point is not to show that but to describe how to fit a STAN model and work with the output. Anyone who wants the full code and data to work with can find it on my website or in Dryad (see the article for a link).

stanMod = """
data{
int<lower = 1> N;
int<lower = 1> J;
vector[N] Temp;
vector[N] y;
int<lower = 1> Curve[N];
vector[J] pctN;
vector[J] pctP;
vector[J] pctH20;
matrix[3, 3] R;
}

parameters{
vector[3] beta[J];
real mu_a;
real mu_b;
real mu_c;
real g_a1;
real g_a2;
real g_a3;
real g_b1;
real g_b2;
real g_b3;
real g_c1;
real g_c2;
real g_c3;
real<lower = 0> sigma[J];
cov_matrix[3] Tau;
}

transformed parameters{
vector[N] y_hat;
vector[N] sd_y;
vector[3] beta_hat[J];
// First, get the predicted value as an exponential curve
// Also make a dummy variable for SD so it can be vectorized
for (n in 1:N){
y_hat[n] <- exp( beta[Curve[n], 1] + beta[Curve[n], 2]*Temp[n] + beta[Curve[n], 3]*pow(Temp[n], 2) );
sd_y[n] <- sigma[Curve[n]];
}
// Next, for each group-level coefficient, include the group-level predictors
for (j in 1:J){
beta_hat[j, 1] <- mu_a + g_a1*pctN[j] + g_a2*pctP[j] + g_a3*pctH20[j];
beta_hat[j, 2] <- mu_b + g_b1*pctN[j] + g_b2*pctP[j] + g_b3*pctH20[j];
beta_hat[j, 3] <- mu_c + g_c1*pctN[j] + g_c2*pctP[j] + g_c3*pctH20[j];
}
}

model{
y ~ normal(y_hat, sd_y);
for (j in 1:J){
beta[j] ~ multi_normal_prec(beta_hat[j], Tau);
}
// PRIORS
mu_a ~ normal(0, 1);
mu_b ~ normal(0, 1);
mu_c ~ normal(0, 1);
g_a1 ~ normal(0, 1);
g_a2 ~ normal(0, 1);
g_a3 ~ normal(0, 1);
g_b1 ~ normal(0, 1);
g_b2 ~ normal(0, 1);
g_b3 ~ normal(0, 1);
g_c1 ~ normal(0, 1);
g_c2 ~ normal(0, 1);
g_c3 ~ normal(0, 1);
sigma ~ uniform(0, 100);
Tau ~ wishart(4, R);
}
"""

# fit the model!
fit = pystan.stan(model_code=stanMod, data=dat,
iter=10000, chains=4, thin = 20)

Not so bad, was it? It’s actually pretty straightforward.

After the model has been run, we work with the output. We can check traceplots of various parameters:

fit.plot(['mu_a', 'mu_b', 'mu_c'])
fit.plot(['g_a1', 'g_a2', 'g_a3'])
fit.plot(['g_b1', 'g_b2', 'g_b3'])
fit.plot(['g_c1', 'g_c2', 'g_c3'])
py.show()

As a brief example, we can extract the overall coefficients and plot them:

mus = fit.extract(['mu_a', 'mu_b', 'mu_c'])
mus = pd.DataFrame({'Intercept' : mus['mu_a'], 'Linear' : mus['mu_b'], 'Quadratic' : mus['mu_c']})

py.plot(mus.median(), range(3), 'ko', ms = 10)
py.hlines(range(3), mus.quantile(0.025), mus.quantile(0.975), 'k')
py.hlines(range(3), mus.quantile(0.1), mus.quantile(0.9), 'k', linewidth = 3)
py.axvline(0, linestyle = 'dashed', color = 'k')
py.xlabel('Median Coefficient Estimate (80 and 95% CI)')
py.yticks(range(3), ['Intercept', 'Exponential', 'Gaussian'])
py.ylim([-0.5, 2.5])
py.title('Overall Coefficients')
py.gca().invert_yaxis()
py.show()

The resulting plot:

We can also make a prediction line with confidence intervals:

#first, define a prediction function
def predFunc(x, v = 1):
yhat = np.exp( x[0] + x[1]*xPred + v*x[2]*xPred**2 )
return pd.Series({'yhat' : yhat})

# next, define a function to return the quantiles at each predicted value
def quantGet(data , q):
quant = []
for i in range(len(xPred)):
val = []
for j in range(len(data)):
val.append( data[j][i] )
quant.append( np.percentile(val, q) )
return quant

# make a vector of temperatures to predict (and convert to the real temperature scale)
xPred = np.linspace(feeding_Final['Temp_Scale'].min(), feeding_Final['Temp_Scale'].max(), 100)
realTemp = xPred * feeding_Final['Temperature'].std() + feeding_Final['Temperature'].mean()

# make predictions for every chain (in overall effects)
ovPred = mus.apply(predFunc, axis = 1)

# get lower and upper quantiles
ovLower = quantGet(ovPred['yhat'], 2.5)
ovLower80 = quantGet(ovPred['yhat'], 10)
ovUpper80 = quantGet(ovPred['yhat'], 90)
ovUpper = quantGet(ovPred['yhat'], 97.5)

# get median predictions
ovPred = predFunc(mus.median())

Then, just plot the median (ovPred) and the quantiles against temperature (realTemp). With just a little effort, you can wind up with something that looks pretty good:

I apologize for only posting part of the code, but the full script is really long. This should serve as a pretty good start for anyone looking to use Python as their Bayesian platform of choice. Anyone interested can get the data and full script from my article or website and give it a try! It’s all publicly available.

## Chart of the Month: Declining extreme weather events in Miami

### Image

Here’s my chart of the month (made using Python). This is a variant of an older post that I cleaned up. I really like the way this turned out. Simple and elegant.

# Why Not? (An Evolution Pictogram)

IF:

Check out David Dogglehoff over there…

AND:

That’s right. Broccoli, cauliflower, cabbage, brussel sprouts, and kale are all one disgusting species.

THEN WHY NOT:

I am aware that a) dogs and the veggies are all one species while the primates here are all different generas and families that diverged millions of years ago (rather than a few thousand), so the differences are much more pronounced and that b) the process of selective breeding (for dogs and veggies) is different than speciation, although it is very similar to sexual selection and I don’t know of any research suggesting that sexual selection is NOT how primates diversified. The common ancestor is an artist’s rendition of Pierolapithecus catalaunicus, which is the suggested common ancestor (or close to it) of humans and great apes.
We are to the great apes what the chinese crested/hairless chihuahua are to dogs

AFTER ALL:

We’re not so different…

Notes: All images from Wikimedia commons except the one of me and the skeleton comparison (which came from Google images).